Chapter 5.4, Problem 22E

To determine

To calculate:

Given system of linear equation has one, infinite or no solution using graph.

Answer to Problem 22E

Solution for given linear equation has one solution.

Explanation of Solution

Given information:

  $\begin{array}{l}2x-2y=16\\ 3x-6y=30\end{array}$

Calculation:

Given equations are:

  $\begin{array}{l}2x-2y=\mathrm{16...}\left(i\right)\\ 3x-6y=\mathrm{30...}\left(ii\right)\end{array}$

After simplifying eq. $\left(i\right)$ ,

  $\begin{array}{l}2x-2y=16\\ -2y=16-2x\end{array}$

Divide by $-2$ on both sides,

  $\begin{array}{l}\frac{-2y}{-2}=\frac{16}{-2}-\frac{2x}{-2}\\ y=-8-x\end{array}$

Another equation is:

  $3x-6y=30$

After simplifying eq. $\left(ii\right)$ ,

  $\begin{array}{l}3x-6y=30\\ -6y=30-3x\end{array}$

Divide by $-6$ on both sides,

  $\begin{array}{l}\frac{-6y}{-6}=\frac{30}{-6}-\frac{3x}{-6}\\ y=-5+3x\end{array}$

For finding out the number of solutions of the system of equation,

The slopes of lines are $-1,3$ respectively for both equations and $y-$ intercept is $-8,-5$

Since the lines are in slope and $y-$ intercept form as $y=mx+c$ , where $m=$ slope of line and $\left(0,b\right)$ be the $y-$ intercept.

As both equations has different slope and different $y-$ intercept thus it represents intersect lines.

So, given equation has one solution.

Thus lines are coincident shown as below: